Well this is an interesting question. For fun, to be a little different
and to limit friction, I would recommend a three-wheel design. Your
vehicle will look a little like a tricycle, with two wheels in back and one
wheel in front. With three wheels intead of four, you'll have a design
that is a little easier and cheaper to build. Also, you'll have a little
less friction than a four-wheel vehicle.

There are two ways to store energy in your elastic band and drive the
wheels of your vehicle. You can twist the band to store energy in it
(which is how many elastic band powered toy airplanes work) or you can
stretch the band to store energy in it. I recommend you use the stretch
strategy. There are a couple of other things to consider about your
elastic band:

You don't want to deliver the energy too fast. If your design stretches
the band too tight and just spins the wheels quickly when you let it go,
the vehicle won't get very far. You want to have your vehicle gather speed
evenly, any energy you waste by spinning your wheels will take off of your
vehicle's total speed.

You don't want the elastic band to be attached in such a way that, after
it delivers the energy, it wraps around the other way and starts to absorb
energy.

Here's what I would suggest:

Design your vehicle, as I said above, as a tricycle. The two REAR wheels,
however, should be larger than the front wheel. Make them about 4 to 8 cm
in diameter and join them firmly to an axle. If the surface you're going
to race on is slick (such as a concrete floor), make sure your rear wheels
have good traction. Rubber wheels would be great, if you can work it out.

The front wheel isn't so important, and it can be smaller. Join the rear
wheels to the front with some sort of frame that lets the rear wheels and
axle turn freely. You will want to lubricate the axles where they rub
against the frame with a drop or two of oil (or even a little Vasoline).
The distance from the front wheel to the rear axle should be about 12-14 cm
(you'll see why in a minute). You should place the egg on the frame
towards the center of the vehicle. Don't let it ride too high, since it
will take away from your overall stability.

To power the vehicle, attach the elastic band to the frame at the front,
just behind the front wheel. Then, run the elastic band back to the rear
axle. If the distance is just right, you will have a little slack in the
band. Take this slack and start wrapping the band around the axle and lap
it over itself. DON'T attach it to the rear axle, just wrap it around.
Once you get the band started wrapping around the rear axle, you can keep
turning the rear wheels to stretch the band and load up energy.

To release the vehicle, you'll put it on the floor and let loose of the
rear wheels. The band will pull on the axle and drive the car. It's
important that the rear wheels are large. If they're small, they'll just
zip around without driving the vehicle. If they're large, they'll resist
turning enough to keep the elastic band from delivering its power all at
once.

As the elastic band unwinds, it will eventually reach the end. Since you
haven't attached it to the rear axle, it will just drag along when it's
done. If you had attached it, it would start to wind up in the other
directions and take away from your speed.

Lastly, you're correct about building the vehicle from balsa wood. With
the exception of the rear wheels, you want it to be as light as possible,
to reduce friction. Still, you might want to look into building it from
pre-fabricated materials. It will be a LOT easier to build from "Tinker
Toys" or "K'Nex" or some other pre-fab parts. Even if you eventually
decide to build it from balsa wood, you can use pre-fab parts to test out
your design and get it right.

Remember: good engineers try out lots of designs and they test, test, test.
If your car has problems at first, you can change your design to fix them.